scholarly journals Numerical Scheme for Finding Roots of Interval-Valued Fuzzy Nonlinear Equation with Application in Optimization

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Ahmed Elmoasry ◽  
Mudassir Shams ◽  
Naveed Yaqoob ◽  
Nasreen Kausar ◽  
Yaé Ulrich Gaba ◽  
...  
Keyword(s):  
Nonlinear Equation ◽  
Numerical Scheme ◽  
Order Of Convergence ◽  
Numerical Test ◽  
Real Life ◽  
Test Problems ◽  
Parametric Form ◽  
Iterative Schemes ◽  
Research Article ◽  
Interval Valued

In this research article, we propose efficient numerical iterative methods for estimating roots of interval-valued trapezoidal fuzzy nonlinear equations. Convergence analysis proves that the order of convergence of numerical schemes is 3. Some real-life applications are considered from optimization as numerical test problems which contain interval-valued trapezoidal fuzzy quantities in parametric form. Numerical illustrations are given to show the dominance efficiency of the newly constructed iterative schemes as compared to existing methods in literature.

scholarly journals A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 904 ◽  
Author(s):  
Afshin Babaei ◽  
Hossein Jafari ◽  
S. Banihashemi
Keyword(s):  
Order Of Convergence ◽  
Additive Noise ◽  
Numerical Solutions ◽  
Numerical Test ◽  
Test Problems ◽  
Heat Equations ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Stochastic Heat Equations ◽  
Collocation Approach

A spectral collocation approach is constructed to solve a class of time-fractional stochastic heat equations (TFSHEs) driven by Brownian motion. Stochastic differential equations with additive noise have an important role in explaining some symmetry phenomena such as symmetry breaking in molecular vibrations. Finding the exact solution of such equations is difficult in many cases. Thus, a collocation method based on sixth-kind Chebyshev polynomials (SKCPs) is introduced to assess their numerical solutions. This collocation approach reduces the considered problem to a system of linear algebraic equations. The convergence and error analysis of the suggested scheme are investigated. In the end, numerical results and the order of convergence are evaluated for some numerical test problems to illustrate the efficiency and robustness of the presented method.

scholarly journals Computer-Based Fuzzy Numerical Method for Solving Engineering and Real-World Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Naila Rafiq ◽  
Naveed Yaqoob ◽  
Nasreen Kausar ◽  
Mudassir Shams ◽  
Nazir Ahmad Mir ◽  
...  
Keyword(s):  
Fuzzy Sets ◽  
Numerical Method ◽  
Nonlinear Equations ◽  
Fuzzy Number ◽  
Fuzzy Numbers ◽  
Numerical Solutions ◽  
Numerical Test ◽  
Real Life ◽  
Triangular Fuzzy Number ◽  
Test Problems

The nonlinear equation is a fundamentally important area of study in mathematics, and the numerical solutions of the nonlinear equations are also an important part of it. Fuzzy sets introduced by Zedeh are an extension of classical sets, which have several applications in engineering, medicine, economics, finance, artificial intelligence, decision-making, and so on. The most special types of fuzzy sets are fuzzy numbers. The important fuzzy numbers are trapezoidal fuzzy and triangular fuzzy numbers, which have several applications. In this research article, we propose an efficient numerical iterative method for estimating roots of fuzzy nonlinear equations, which are based on the special type of fuzzy number called triangular fuzzy number. Convergence analysis proves that the order of convergence of the numerical method is three. Some real-life applications are considered as numerical test problems from engineering, which contain fuzzy quantities in the parametric form. Engineering models include fractional conversion of nitrogen-hydrogen feed into ammonia and Van der Waal’s equation for calculating the volume and pressure of a gas and motion of the object under constant force of gravity. Numerical illustrations are given to show the dominance efficiency of the newly constructed iterative schemes as compared to existing methods in the literature.

An Optimal Eighth-Order Scheme for Multiple Zeros of Univariate Functions

2019 ◽  
Vol 16 (04) ◽  
pp. 1843002 ◽  
Author(s):  
Ramandeep Behl ◽  
Fiza Zafar ◽  
Ali Saleh Alshormani ◽  
Moin-Ud-Din Junjua ◽  
Nusrat Yasmin
Keyword(s):  
Order Of Convergence ◽  
Real Life ◽  
Eighth Order ◽  
Multiple Zeros ◽  
Iterative Schemes ◽  
Life Problems ◽  
Order Scheme ◽  
Multiplicity Formula ◽  
First Time ◽  
Better Than

We construct an optimal eighth-order scheme which will work for multiple zeros with multiplicity [Formula: see text], for the first time. Earlier, the maximum convergence order of multi-point iterative schemes was six for multiple zeros in the available literature. So, the main contribution of this study is to present a new higher-order and as well as optimal scheme for multiple zeros for the first time. In addition, we present an extensive convergence analysis with the main theorem which confirms theoretically eighth-order convergence of the proposed scheme. Moreover, we consider several real life problems which contain simple as well as multiple zeros in order to compare with the existing robust iterative schemes. Finally, we conclude on the basis of obtained numerical results that our iterative methods perform far better than the existing methods in terms of residual error, computational order of convergence and difference between the two consecutive iterations.

scholarly journals Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1242
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Eulalia Martínez ◽  
Majed Aali Alsulami
Keyword(s):  
Iterative Methods ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Real Life ◽  
Multiple Roots ◽  
Computational Results ◽  
Nonlinear Functions ◽  
First Order ◽  
Life Problems ◽  
Derivative Free

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.

The Numerical Analysis and Simulation of a Linearized Crank-Nicolson H1-GMFEM for Nonlinear Coupled BBM Equations

2014 ◽  
Vol 513-517 ◽  
pp. 1919-1926 ◽  
Author(s):  
Min Zhang ◽  
Zu Deng Yu ◽  
Yang Liu ◽  
Hong Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Scheme ◽  
Mixed Finite Element ◽  
A Priori ◽  
Numerical Test ◽  
Mixed Finite Element Method ◽  
Spatial Direction ◽  
Time Direction ◽  
Crank Nicolson

In this article, the numerical scheme of a linearized Crank-Nicolson (C-N) method based on H1-Galerkin mixed finite element method (H1-GMFEM) is studied and analyzed for nonlinear coupled BBM equations. In this method, the spatial direction is approximated by an H1-GMFEM and the time direction is discretized by a linearized Crank-Nicolson method. Some optimal a priori error results are derived for four important variables. For conforming the theoretical analysis, a numerical test is presented.

Chemical equilibrium systems as numerical test problems

1990 ◽  
Vol 16 (2) ◽  
pp. 143-151 ◽  
Author(s):  
Keith Meintjes ◽  
Alexander P. Morgan
Keyword(s):  
Chemical Equilibrium ◽  
Numerical Test ◽  
Test Problems

scholarly journals Decision making under interval uncertainty: toward (somewhat) more convincing justifications for Hurwicz optimism-pessimism approach

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Warattaya Chinnakum ◽  
Laura Berrout Ramos ◽  
Olugbenga Iyiola ◽  
Vladik Kreinovich
Keyword(s):  
Decision Making ◽  
Real Life ◽  
Interval Uncertainty ◽  
Lower And Upper Bounds ◽  
Worst Case ◽  
Content Type ◽  
Expected Gain ◽  
Utility Approach ◽  
Interval Valued ◽  
Weighted Combination

Purpose In real life, we only know the consequences of each possible action with some uncertainty. A typical example is interval uncertainty, when we only know the lower and upper bounds on the expected gain. A usual way to compare such interval-valued alternatives is to use the optimism–pessimism criterion developed by Nobelist Leo Hurwicz. In this approach, a weighted combination of the worst-case and the best-case gains is maximized. There exist several justifications for this criterion; however, some of the assumptions behind these justifications are not 100% convincing. The purpose of this paper is to find a more convincing explanation. Design/methodology/approach The authors used utility approach to decision-making. Findings The authors proposed new, hopefully more convincing, justifications for Hurwicz’s approach. Originality/value This is a new, more intuitive explanation of Hurwicz’s approach to decision-making under interval uncertainty.

scholarly journals Splitting-particle methods for structured population models: Convergence and applications

2014 ◽  
Vol 24 (11) ◽  
pp. 2171-2197 ◽  
Author(s):  
J. A. Carrillo ◽  
P. Gwiazda ◽  
A. Ulikowska
Keyword(s):  
Numerical Scheme ◽  
Order Of Convergence ◽  
Operator Splitting ◽  
Population Models ◽  
Particle Simulation ◽  
Particle Methods ◽  
Test Cases ◽  
Structured Population Models ◽  
Convergence Error ◽  
Structured Population

We propose a new numerical scheme designed for a wide class of structured population models based on the idea of operator splitting and particle approximations. This scheme is related to the Escalator Boxcar Train (EBT) method commonly used in biology, which is in essence an analogue of particle methods used in physics. Our method exploits the split-up technique, thanks to which the transport step and the nonlocal integral terms in the equation can be separately considered. The order of convergence of the proposed method is obtained in the natural space of finite non-negative Radon measures equipped with the flat metric. This convergence is studied even adding reconstruction and approximation steps in the particle simulation to keep the number of approximation particles under control. We validate our scheme in several test cases showing the theoretical convergence error. Finally, we use the scheme in situations in which the EBT method does not apply showing the flexibility of this new method to cope with the different terms in general structured population models.

A Novel Python Toolbox for Single and Interval-Valued Neutrosophic Matrices

2020 ◽  
pp. 281-330
Author(s):  
Said Broumi ◽  
Selçuk Topal ◽  
Assia Bakali ◽  
Mohamed Talea ◽  
Florentin Smarandache
Keyword(s):  
Real Life ◽  
Neutrosophic Sets ◽  
Vague Set ◽  
Life Problems ◽  
Consistent Information ◽  
Interval Valued

Recently, single valued neutrosophic sets and interval valued neutrosophic sets have received great attention among the scholars and have been applied in many applications. These two concepts handle the indeterminacy and consistent information existing in real-life problems. In this chapter, a new Python toolbox is proposed under neutrosophic environment, which consists of some Python code for single valued neutrosophic matrices and interval valued neutrosophic matrices. Some definitions of interval neutrosophic vague set such as union, complement, and intersection are presented. Furthermore, the related examples are included.

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